807
The Electronic Structure of Methylene'" James F. Harrisonlb and Leland C. Allen
Contribution from the Department of Chemistry, Princeton Uiiirersitj?, Princeton, New Jersej 08540. Receiued September 11, 1968 Abstract: Ab initio valence-bond wave functions are reported for the 3B1, 'A1, and 'B1 states of CH, as a function of angle with R(C-H) = 2.00 bohrs. Up to 48 valence-bond structures for each state were used to expand the variational wave functions (each VB Ftructure is a single term of an expansion which is composed of those appropriate linear combinations of determinants required to assure a definite spin and space symmetry). A nonorthog-
onal free atom basis constructed with Gaussian lobe functions was employed. Calculations with scaled hydrogen orbitals were likewise carried out. The total energy of the ground state, 3B1,near its equilibrium position is ET (140") = -38.9151 Hartree units. Ab initio MO-SCF wave functions were also computed as a function of angle for the 'A1 state, and MO wave functions for the lS3B1states were constructed from single excitations of the MO 'Al state. The order of states is computed to be 3 B < ~ 'AI < 'BI with large equilibrium angle for 1.3B1,108" for 'Al, and an IA1-lB1 energy separation of 1.52 eV. Spectroscopic data are available for these properties, and the agreement between theory and experiment is quite good. Larmor diamagnetic susceptibility terms are computed to be - 19.75 x 10-6 and - 19.57 X erg,'(G2 mole) for the 'A1 and 'B1 states, respectively, compared to an estimated The 3 B heat ~ of atomization is calculated as 6.36 eV (experimental value 8.5 experimental value of - 12 x eV). Expectation values of the following quantities (for which experiments are currently lacking) have been obtained as a function of angle: dipole moments, quadrupole tensor, diamagnetic contribution to the nuclear magnetic shielding constant of the protons, diamagnetic anisotropy, electric field gradient tensor, quadrupole coupling constants for deuterated methylene, ( l / r H ) , (l/rc>,heats of atomization of the 'A, and IB1 states, the 3B1-'A1 energy separation, and oscillator strength for singlet-singlet transitions. Considerable attention has been given to detailed descriptions of the charge distributions and to the implications of our results for divalent carbon chemistry. There is a long and intricate history of experimental and theoretical attempts to elucidate the electronic structure of methylene, and in order to aid over-all understanding of this molecule, a rather extensive review and analysis of previous work has been included.
I. Introduction ethylene is the basic unit for divalent carbon chemistry and is one of the most important molecules in chemistry. The existence of a triplet ground state and relatively low-lying excited singlet state in CH2 has led to a unique and rapidly developing branch of organic chemistry. It is challenging, and perhaps surprising, that spectroscopic experiments on this comparatively simple species have been very difficult and often ambiguous. Similarly, there has been a long sustained and rather unsuccessful history of theoretical effort. The theoretical electronic structure study presented here is an ab initio valence-bond treatment of methylene's ground and low-lying excited states. F o r constructing various spin states in an eight-electron, threecenter system such as CH2, the valence-bond approach is particularly suitable for generating high accuracy, chemically interpretable wave functions. For coinparison, an ab initio self-consistent field molecular orbital solution was also obtained for one of the states. In view of the history and importance of this species, it is satisfying that the present study is in general agreement with the best existing experimental work and that experimental and theoretical results now may be treated with roughly equal confidence. A variety of other properties, not presently available experimentally, also has been predicted. In the next section the wave-function formulation
M
(1) (a) This research was supported in part by the Chemistry Section of the National Science Foundation, Grant No. NSF-GP-8907, and the Directorate of Chemical Sciences of the Air Force OEce of Scientific Research, Contract No. A F 49(638)-1625. (b) Author to whom inquirie5 should be addressed at Department of Chemistry, Michgan State University, East Lansing, Mich. 48823.
and basis set determination is given. Section IT1 presents, and compares with experiment, results on the order of states, geometries, binding energies, and transition probabilities. Section IV predicts some one-electron properties, most of which have not been obtained experimentally. Section V is an extensive review and analysis of prior theoretical work and a brief outline of the experimental situation. Taken together with a review by Gaspar and Hammond? this provides the background which, because of it5 rich and intricate history, is particularly necessary and appropriate to an understanding of methylene.
11. Method of Calculation A. Description of Technique. Most of the energy calculations carried out in this study have employed the valence-bond (VB) formalism taking into account fully the nonorthogonality of the atomic basis. This was accomplished by using the Lowdin formulation for performing linear variational calculations over a basis of Slater determinants, the elements of which are nonorthogonal. A digital computer program for this procedure has been written by ErdahL4 The program is capable of solving the linear variational problem over a basis of 80 valence-bond structures each of which may be a linear combination of eight Slater determinants (each determinant in turn consisting of up to 32 spin orbitals). We have occasion to compare the results of a valence-bond calculation with an LCAO-MO calculation over the same basis. The M O results have (2) P. P. Gaspar and G. S . Hammond, "Carbene Chamistry," W . Kirmse, Ed., Academic Press, New York, N. Y., 1964, pp 235-273. (3) P. 0. Lowdin, Ph.vs. Rec., 97, 1474, 1490, 1509 (1955). (4) R. M. Erdahl, Ph.D. Thesis, Princeton University, 1966
Harrison, Allen
1 Electronic Structure of Methylene
808 Table 1. Atomic Basis and Primary Structures Used in Constructing VB-Doubly Occupied 1s and 2s Atomic basis 1. 2. 3. 4. 5. 6. 7.
Carbon 1s Carbon 2s Hydrogen 1 1s Carbon pE Hydrogen 1 1s Carbon p. Carbonp,
+ hydrogen2 -
1s
hydrogen2 1s
1) 2) 3) 4) 5) 6) 7) 8) 9) 10)
1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2 2
3 3 3 3 4 4 4 5 5 6
4 5 6 7 5 6 7 6 7 7
been obtained via the Hartree-Fock-Roothaan equations. All integrals were generated over Gaussian lobe functions and then transformed to the basis being used in the calculation. For example, in carrying out a VB calculation over an atomic orbital basis, the integrals were evaluated over the Gaussians and then transformed into the atomic orbital b a s k 5 The atomic orbitals are close t o atomic Hartree-Fock solutions. The notation used for the Slater determinant has been described previously4 and will only be briefly outlined here. We restrict ourselves t o systems containing an even number of electrons and consider only deterniinants having the number of LY spins equal to the number of /3 spins. This requires that we consider the S, = 0 component of triplets, etc. We collect all of the spatial orbitals associated with the same spin and call this ordered set of spatial orbitals a primary structure. Each determinant requires for its definition two primary structures, one to define the a-spin configuration and one to define the @-spinconfiguration. A linear combination of determinants which is an eigenfunction of the spin operator and is of definite spatial symmetry is called a valence-bond structure. The linear variation problem is then solved over a basis of valence-bond structures. B. Basis Functions. We use the full nonrelativistic Hamiltonian in atomic units
and we introduce the atomic basis presented in Table I. In C2, the functions 1, 2, 3, and 6 transform as al, 4 and 5 as b,, and 7 as bl. We take the x axis as the Cz axis so that in the linear configuration the z axis is the internuclear line (see Figure 1). Throughout this study we take the CH distance as 2.00 bohrs. C. Valence-Bond Considerations. We have carried out three series of calculations employing the VB formalism. (a) VB I. In the first VB calculation we consider only determinants having the carbon configuration ( 1 ~ ) ~ ( 2 s;) *i.e., we do not allow for hybridization of the carbon 2s and 2p orbitals. Under these conditions there are of the eight electrons in the problem only four “active.” The number of primary structures which arise from the permutation of these four electrons among the atomic orbitals 3 through 7 is (E) = 10, and from these 10 primary structures (shown in Table I) ( 5 ) J. L. Whitten and
ix
Primary structures
L. C.Allen, J . Chem. Phys., 43, S170 (1965);
J. L. Whitten, ibid., 44,359 (1966).
Journal of the American Chemical Society
/ 91:4 /
Figure 1. Coordinate system,
we may form 100 unique 8 X 8 determinants. We then use projection operators to project the space spanned by these determinants into singlet and triplet subspaces. Then within the singlet and triplet spaces we project out all functions of symmetry AI, BI, and Bz. The VB structures so generated are displayed in Table 11. The linear variational problem was solved within each space of definite symmetry and spin. The potential curves which were generated are shown in Figure 2. The energies used to construct these curves are listed in Table 111. The wave functions are not presented but are available upon request. We note that all states correlate with the proper linear symmetry, and we present in Table IV the energies of the various states of linear CH, relative to the 32,- state. Since we have employed only valence-state orbitals, the states with rather high energies have little physical meaning. (b) VB 11. In the second calculation we consider those determinants having the carbon configuration (Is)*; i e . , we allow completely for hybridization between the carbon 2s and 2p. Under these conditions there are six “active” electrons which may be permuted among the six spatial orbitals 2 through 7 . This gives rise to (3 = 20 primary structures (see Table V) from which we may form 400 8 x 8 determinants. The space defined by these determinants was partitioned into singlet, triplet, and quintuplet subspaces. From the singlet space we projected the ‘A1 and IB1 functions, while from the triplet space we projected only the 3B1. We did not consider the quintuplet subspaces, and because of digital computer limitations we deleted from consideration primary structures 1 and 20. Physically, this is not a significant restriction. The 18 structures employed are listed in Table V. VB structures constructed from this set are shown in Table VI. The linear variation problem for the states 3B1,’AI, and ‘Bi was solved for several values of the H-C-H angle, and the results are presented in Figure 3 as curves 3B1(II), lAl(Il), and lB1(II). The energies used in constructing these curves are presented in Table VII. (Wave functions are not tabulated but are available upon request.) We note that this rather extensive calculation predicts the 3B1lowest at all angles while calculation VB I predicted that the multiplicity of the ground state would change from triplet to singlet as the molecule is bent from the linear configuration. (c) VB 111. In the third calculation we use exactly the same VB structures as in calculation 11. However, we modify the atomic basis slightly by scaling the hydrogen atom exponents by 1.8 (effective exponential H scaling of This scaling factor, or values close to it, has been found optimum for hydrocarbons i n this and several other laboratories. The results are
February 12, 1969
dm).
Table 111. Energy" us. Angle for Methylene VB-Doubly
e,
5
Occupied Is and 2s (RCH= 2.00 Bohrs)
deg
'Ai
'A2
'BI
'Bz
80 100 120 140 160 180
-38.7474 -38.7559 -38.7355 -38.6977 -38.6593 -38.6437
-38.5693 -38.5347 -38.4789 -38.4105 -38.3379 -38.2915
-38.5819 -38.6382 -38.6618 -38.6642 -38.6547 -38.6477
-38.2780 -38.3052 -38.3109 -38.3052 -38.2960 -38.2915
3A1 -38.2483 -38.2642 -38.3283 -38.3943 -38.4397 -38.4555
3Az
932
-38.6021 -38.5671 -38.5118 -38.4441 -38.3724 -38.3275
-38.6266 -38.6860 -38.7122 -38.7177 -38.7119 -38.7072
-38.3926 -38.4096 -38.4015 -38.3783 -38.5471
...
Energy in Hartree units.
Harrison, Alien
/ Electronic Structure
of Methylene
810
I
Table IV.
*ye VB-Doubly Linear
Energy Spectrum of Linear Methylene Occupied Is and 2s
-
Bent"
-
3B~
'A,
'&+
3 A"
IAU
3Az
'
*g
35,-
3A11 I&3XUf
1s" lZ"+ 3&
7.-
" -38.600b
'A1
lB1
3A,, "BI 'Ai, 'Bi 3As, 3B2 'A*, 'nz 3A2 'A*, 3B1 LA? 3B2 '€32, 'A? 1B2 3A1, 3 B ~
I&-
'€31
32,f
3A1
- 38.7072 - 38.6473 - 38.5909 -38.4555 - 38.4235 - 38.3275 - 38.2915 - 3E. 2375 - 38.2156 - 38,2005 -38 1592 -38.0063 - 37.9489 - 37.7168 - 37,7017 - 37.6598
AE, eV
~
3&-
~
Energyb ~~~~~
0.0000 1.6299 4.1645 6.8488 7.7195 10.3316 11.3112 12.7E05 13.3764 13, '7873 14.9111 19,0715 20.6333 26.9488 27.3597 28.4998
a Bent molecules states correlated with linear symmetries. ergy in Hartree units.
-38.80Ol.-I I80
I60
u
I
I
I40
I 20
100
80
4-0Figure 2. Energy us. angle for various states of CH?. VB-doubly occupied Is and 2s.
I - B E S T ATOM BASIS-IS AND 2s DOUBLY OCCUPIED I I - B E S T ATOM BASIS- 1s DOUBLY OCCUPIED E I - S C A L E D HYDROGEN 1s DOUBLY I OCCUPIED
-38,5801
/ ~
-38.620b
Table V. Atomic Basis and Primary Structures Used in Both VB-Doubly Occupied Is Only and VB-Doubly Occupied 1s with Scaled Hydrogen Atomic Basis 1) Carbon 1s 2) Carbon 2s hydrogen 2 Is 3) Hydrogen 1 1s 4) Carbon pr 5) Carbon pz 6) Hydrogen 1 1s - hydrogen 2 1s 7) Carbon pu Complete Set of Primary Primary Structures Actually Structures for Basis Orbitals Used Employed in Calculationa 1 ) 1 2 3 4 1 1 1 2 5 6 2 ) 1 2 3 5 2 1 1 3 5 6 5 6 4 3 ) 1 2 3 6 3 ) l 4 ) 1 2 3 7 4 ) 1 2 3 7 5 ) 1 2 4 5 5 ) 1 2 4 7 6 ) 1 2 4 6 6 ) 1 3 4 7 7 7 ) 1 2 4 7 7 ) 1 2 5 8 ) 1 2 5 6 8 ) 1 2 6 7 9 ) 1 2 5 7 9 ) 1 3 5 7 1 2 6 7 10) 1 3 6 7 10) 11) 1 3 4 5 llj 1 5 4 7 1 3 4 6 12) 1 6 4 7 12) 13) 1 3 4 7 13) 1 2 3 5 14) 1 3 5 6 14) 1 2 3 0 15) 1 3 5 7 15) 1 2 5 4 1 3 6 7 16) 1 2 6 4 16) 1 3 5 4 1 7 j 1 4 5 6 17) 4 5 7 18) 1 3 6 4 18) 1 1 4 6 7 19) 6 7 20) 1 5
+
a
w
I
/
-38.8201 'Ag (EI)
/
'Bl(IU)
I
i1
-38 900, *-• 3Zg(JIr I) -*
-38 940 I80
-
*/
'
381(m) * / *
I I
150
I
140
I20
+@--
U loo
Figure 3. Energy us. angle for the dB1,lBl, and 'A1 states of CH?. A comparison of three levels of VB calculations.
Journal of the American Chemical Society
shown in Figure 3 as curves %(ILI), IAl(I1I), and lBl(L1I). The energy values and wave functions used to construct these curves are given in Tables VI1 and VIII. resaectivelv. We see that in relation t o calculation ' I1 {he eneigy has improved, but the qualitative features are unchanged. D. Molecular Orbital Considerations. We obtained the single determinant solution to the Hartree-FockRoothaan equations for the 'A1 state. The solution was obtained as a function of angle using the carbon group orbital basis5 with scaled hydrogen exponcnts. The scaling factor ( q 2 =: 1.8) is the same as was used in VB 111. The one-electron energies are plotted as a function of angle in Figure 4. We note the general
1 , : ' /
e
1:
Structures have been renumbered.
/
'131x9 == (E) -:q:-* *-*'A&n) I '~'(m) -38.860 1
En-
91:4
/ February 12, I949
811 Table VI. Valence-Bond Structures in Primary Structure Notation Used in Both VB-Doubly VB-Doubly Occupied 1s with Scaled Hydrogen
Occupied 1s Only and
Singlet A1 State 25)
agreement of 3al and 2b2 curves with those presented by Walsh.6 The 2al MO has the form
+
+
+
a(hl h2) P(l2s) Alp,)) with a,p , X > 0. This orbital is very much involved in the binding. (The lal molecular orbital is essentially the carbon Is and is chemically uninteresting,) In the (2ad
=
(6) A. D. Walsh, J . Chem. Soc., 2260 (1953).
linear configuration, X = 0. As the molecule is bent the carbon sp hybrid interacts with the symmetric combination of hydrogen orbitals causing electron density t o be built up in the region between the carbon and hydrogen atoms. As the molecule is bent from 180 t o 90°, this orbital becomes more stable. The lbz orbital has the form (with 6 and E > 0) llb2)
=
81/22 - h l ) -
Harrison, Allen
EIP, )
Electronic Structure of Methylene
812 Table VII.
Energy"
VS.
Angle for Methylene VB-Doubly
0, deg
Free atom hydrogen
80 100 120 140 160 180
- 38,7871 - 38.8243 - 38,8382 - 38.8395 -38.9354 - 38.8329
Occupied 1s Only
-
r-
3B1 S
'Ai Free atom hydrogen
Scaled hydrogen
Free atom hydrogen
- 38,8505
- 38,7736 - 38.7898 - 38,7843 -38.7712 - 38.7625
- 38.8429 - 38,8643 - 38,8622 - 38.8491 -38.8395
...
...
- 38.6875 - 38.7321 - 38.7518 - 38,7580 - 38.7576 - 38.7565
-38,8939 - 38.9123 -38.9151 -38,9105 - 38.9075
'B1 ___ Scaled hydrogen
7
caled hydrogen
- 38,7471 -38.7988 - 38.8239 -38.8322 -38.8318 -38.8304
Energy in Hartree units.
This orbital has a node along the C2 axis and becomes less stable as the molecule is bent from 180 t o 90". Since a charge density contour cannot pass across the nodal plane containing the Cz axis, the electrons in this orbital are essentially localized in the regions of the CH bonds, and the destabilization of the orbital upon bending may be interpreted as bond-bond repulsion. The 3al orbital is the highest occupied MO for the 'A1 state. This orbital has the form
+ h l ) + K(/lS) - plpZ))
(381) = -7jh2
with 7, K, p > 0. I n the linear configuration this MO is a pure carbon p, orbital and is nonbonding. As the molecule is bent from the linear configuration, the hydrogen atoms experience a net attraction due to the symmetric combination of the 1s orbitals. As the molecule is bent the orbital becomes more stabilized. The 3al orbital may be thought of as describing a hydrogen molecule interacting with a sp-hybridized carbon atom, the sp hybridization being such that the maximum of the electron density is along the --x axis. The first virtual orbital is the lbl which is the pry orbital of carbon. The single determinant 'A1 state is defined by the configuration (la1)2(2al)2( 1b2)z(3a1)2. Single excitation
into the lbl level yields the configuration (la1)?(2a1)*(lbz)2(3al)1(lbl)1,from which we may construct the 3B1and lB1 states. It is expedient to write these MO functions in the primary structure notation. We therefore define Primary structure
Orbital characterization
1
l a l 2al 3al Ib,
2
l a l 2al lb2 lbl
In this notation we have 1'Ai)
I'B1)
=
=
(1,l)
(1,2)
(1)
+ (2,1)
(2)
I3Bd = (1,2) - (2,1) (3) The energy as calculated with these functions is presented in Table IX and plotted in Figure 5 along with the results of calculation VB I11 for comparison. Particularly interesting is the similarity in the curvature of the variationally determined VB I11 energy curves obtained by using virtual orbitals. Since the carbon bl MO is the p, orbital, its atomic character is maintained inviolate for all angles. Excitation into this first virtual orbital is therefore rather unique, and one would not expect the parallelism between the VB determined curve
-38.7201
0.lOr
0.00-
-38.740
L ~
-0.10 I ~
'A,
-0.20~
-
STATE
-38.7601
v, -38.780
t
5 -38.800 0
5
-38.820
0
k
5 -38.840 >-
I
1
.-
E -38.860 w
z
-38.880L
-0.eot
- 38.900k -38.9201
,
,i
,
,
,
160
140
120
I00
80
-38.940
!
80
180 I
100
I
120
I
140
I
160
0Figure 4. One-electron energies of the 'A1 state of CH2 us. angle (Walsh diagram).
Journal of the American Chemical Society / 91:4
-8 Figure 5. Energy us. angle for the 3B1, IB1, and 'A, states of CHI. The VB calculation has the Is doubly occupied and scaled hydrogens (7 = 6 8 1 .
/ February 12, 1969
813 Table VIII. Valence-Bond Wave Functions for Methylene (Doubly Occupied Is and Scaled Hydrogen) VB struct
e = 120"
e = 140'
0,098631 0.014225 0.047998 - 0.027802 -0,035915 0,02297 1 -0.077863 0.012538 -0.027095 0.033100 0.008144 0.004278 -0.062866 -0.092884 0.001255 - 0,048088 - 0.021 983 - 0.001935 -0,061277 -0.045958 -0,001764
0.080754 0,009094 0,044390 -0.022158 -0.043865 0.021100 -0,084385 0.008412 -0,033564 0.030519 0.009080 0,002750 -0,051708 -0,081066 0,000929 -0,041393 -0,020977 -0.001385 - 0,048352 -0,034409 -0.001196
0.062220 0.006979 0,035890 -0.016633 -0,051084 0 . 016655 -0.089860 0.005255 -0,043466 0.025430 0.010871 0.001283 -0.044691 -0.063635 0.000606 - 0.032815 -0,017832
-0.032938 0.004105 -0,008481 0,025418 -0,052313 0.010635 -0.093542 0.019022 0.002978 - 0.0001 36 -0.015528 0.015939 0.004835 -0.003027 -0.00611 1 0,010840 0.011038 0.010405 0.016602 0.123536 0.027216 0,041641 -0,007367 0.006266
-0.018136 0,001170 - 0.014054 0.016641 -0,056330 0.015420 -0.094116 0.029504 0.003122 0.000366 -0.026924 0.017618 0,006303 - 0.003063 - 0,005631 0.012429 0.010878 0.008589 0.012028 0.091 872 0.023345 0.036584 -0.00623 5 0.005279
- 0 .006679
O.oooO75 -0.024885 0.007805 -0.057047 0,025653 -0.090522 0.049641 0.002686 0.000340 - 0,040077 0.025724 0,008125 - 0,004526 - 0 ,002891 0.013035 0.008660 0,005 171 0.005602 0.062458 0,018867 0.026088 -0,004865 0.003850
-0.ooO303 0,000044 -0,042062 0,000943 -0,049730 0.042839 -0.077663 0.081184 0.001123 0,000063 -0,046193 0.041917 0.008782 -0.007546 -0,002816 0.011786 0,002317 -0,000195 0.002189 0,028669 0,009664 0.011792 -0,002455 0,001802
0,095722 0.003792 0.038151 -0,028329 0.011784 - 0.015705 -0,030122 - 0,024578 0.036648 0,070032 0.073192 0.031963 -0.129937 - 0.024013
0.074151 0.002553 0.034490 -0.022404 0,016669 -0,013683 -0,022550 - 0,02371 6 0.038422 0.062101 0.081220 0.026681 -0.113809 -0.019244
0,054363 0.003355 0.026331 -0.015906 0,022543 -0.010383 - 0.013554 -0.018836 0.036712 0,049118 0,090288 0.019301 -0,091477 -0.013632
0.030154 0.002948 0,014406 -0,008414 0.027792 -0,005654 -0.006787 -0,010442 0.033752 0.028183 0,098546 0.010254 -0,053890 -0.007181
e
= 100"
e
= 160"
VB struct
e = 120"
6
-0.020914 -0.036816 0.000597 -0.127348 0.077361 -0.018054 -0.029877 0.019607 0.002145 0.166649 -0.141748 0.001615 0.004044 0.002698 0.067891 - 0,008460 0.020083 0.125665 -0.122593 0.022399 0.328561
-0.018669 - 0.031609 0.000334 -0.120756 0.088256 -0.015297 -0.024638 0.017920 0,001500 0.165705 -0.148902 0.001167 0.005747 O.OOO640 0.068136 - 0.008013 0.020233 0.125956 -0.128754 0.020159 0.346797
-0,014765 -0.024313 0.000160 -0.120793 0.103503 -0.012094 -0.017878 0.014521 0.000997 0.164154 -0.155198 0.000826 0.006997 -0.003994 0.070219 -0,007785 0.020605 0.128711 -0.137599 0.018545 0,371916
- 0.008474
0.006040 0.012378 -0.006693 0.222356 -0,105902 -0.046260 0.038801 -0,003613 -0,031410 0.005311 -0.005381 0.004761 0.011229 - 0.002231 -0.004769 -0.034636 - 0.004584 -0.032495 -0.251810 -0.011190 0.201407 -0.065287 0.014557 0.055699
0.006339 0.010487 - 0.005017 0.243448 -0,127126 -0.068585 0.050959 -0.005115 -0.028516 0,006374 -0.004641 0.004275 0.008642 0.002293 - 0,006029 - 0.030237 - 0.003628 - 0.031145 -0.206945 - 0.010909 0.236463 - 0.089105 0.016602 0.049808
0.006331 0.006601 -0.003468 0.253586 -0.148555 -0.114874 0.079775 -0.006421 - 0.021767 0.007364 -0.003797 0.002952 0.005533 0.007208 -0,007485 - 0,026102 - 0.001829 -0,023991 -0.148327 -0.009502 0.265543 -0.139339 0.018469 0.040515
0.004592 0.001973 -0.001608 0.228921 -0.147597 -0.191384 0,128475 -0,005682 - 0.013822 0.007767 -0.002264 -0.000477 0.002288 0.009637 -0,O10095 -0,016819 -0.000094 - 0.006356 - 0,069403 -0.004876 0,255844 - 0.221027 0.015100 0.023079
-0,039904 0.003363 -0.109390 0.006567 -0.002821 -0.004043 -0.033179 -0.299730 -0.009985 -0.171894 0.002?19 0.430061 0.001660 0.152913
- 0.030440 -0.016828
e
=
100"
=
1400
e = 1600
aBl State 1
2 3
4 5 6 7 8
9 10
11 12 13 14 15 16 17 18 19 20 21
0.035925 0.004352 0,020955 -0.009528 -0,057285 0,009472 -0.095576 0,002689 -0,053486 0.015270 0.012900 O.OOO368 -0,040320 -0,036559 0,000304 -0,019192 -0,010849 -0.000880 -0.000432 -0,034094 -0,018084 -0.022857 -0,011544 -0.000719 -0,000339
22 23 24 25 26 21 28 29 30 31 32 33 54 35 36 37 38 39 40 41 42
-0.013761 0.000061 -0.123568 0.118484 - 0.009479 - 0.012012 0.011087 0.000702 0,163788 -0,161192 0.000626 0.007560 -0.006624 0.073018 - 0.007772 0.021181 0.132456 -0.146768 0.017818 0.397381
'A1 State
1 2 3 4 5
6 7 8
9 10
11 12 13 14 15 16 17 18
19 20 21 22 23 24
25 26 21 28 29 30 41 32 33 34 35 36 37 38 39 40 41 42 43 44 55
46 47 48
IBI State
1 2 3 4 5 6 7 8
9 10 11 12 13 14
and that obtained by virtual excitations t o be general. The single determinant solution for the 'Al state becomes a very poor representation of this state as this molecule approaches a linear configuration. This is anticipated by noting that the 'A1 state must correlate with the 'Ag state in the linear configuration, and a single determinant cannot represent the 'Ag state. In the linear configuration the functions (1,l) and (2,2) are
15 16 17 18 19 20 21 22 23 24 25 24 27 28
-0,046540 0.005273 -0.135074 0.008313 -0,003191 -0,005342 -0.054168 -0.265526 -0.015128 -0.159033 0.003112 0.400567 0.002192 0.136716
0.001963
0.000901
- 0.078279 -0.041508
0.004532
0.002323
-0,002909 -0.002161 -0,002712 - 0 , O O1375 -0.016695 - 0.004692 -0.340419 -0.377673 -0,005108 -0.001418 -0.182241 -0.190616 0.001279 0.000368 0,460179 0.487836 0,000953 0.000287 0.171098 0.186850
degenerate since the 3al + irz and l b l + 7ru as 0 -t 180'. Therefore, the 'A1 state is better represented as a mixture of (1,l) and (2,2) at all angles. We therefore formed
/'AI*)
=
sin X (1,l)
+ cos X (2,2)
(4)
and varied X at each angle 8. The energy as calculated with this function is shown in Table IX and plotted us. Harrison, Allen
1 Electronic Structure of Methylene
814 Table IX. Total Energy" DS. Angle for Methylene Molecular Orbital Calculation
-0,
deg 80 100 120 140 160
'Al* -38.8007 -38.8274 -38.8275 -38.8137 -38.7986
1A,*c
-38.8131 -38.8401 -38.8418 -38.8316 -38.8232
3B1d
'Bld
-38.7199 -38.7782 -38.8083 -38.8204 -38.8224
AH, for the first three states of CH2 are defined by the reactions (the precision of our calculations does not require distinction between energy and enthalpy)
+ 2H(*S) + AH( 3B1) CHd'AJ +C('D) + 2H('S) + AH('A1) CHd'Bl) +C('D) + 2H('S) + AH('B1) CH,( 3B1) +CPP)
-38,8242 -38.8711 -38.8907 -38.8922 -38,8849
Single determinant SCF solution a Energy in Hartree units. (lal)2(2al)'(lbi)2(3a,)2. Variation calculation over two determinant bases, consisting of (la1)2(2a1)2(lbr)?(3al)~ and (la1)2(2a1)2(lb2)2(lb1)2. States are formed by a single excitation from the 3aI MO to the lbl virtual orbital.
(A) (B) (C)
Experimental estimates for AH( 3B1)have been obtained by considering the process C H Q ( ~ B+ ~ ) CH(%)
+ H('S) + x
and from spectroscopy$ we know CH(+)
--+C(3P) + H(2S) - 3.47 eV
angle in Figure 5. It indeed behaves properly, becoming degenerate with the 'B1, as the linear configuration is approached.
Therefore
111. Discussion of Energy Results A. Order of States and Geometry. We present in Table X the equilibrium angles and energies of the "1, 'Al. and 'B1 states as predicted from the calculations we have described and for comparison the results of Foster and Boys.7 We note the following. (a) Every calculation (with the exception of VB I) predicts the order of the states as 3B1< 'A1 < lB1 (VB I predicts 'A1 < 3B1< IBI). (b) Every calculation (with the exception of VB I) predicts the 3B1 to be the ground state for all angles considered (VB I predicts that in going from the linear to the bent configuration the ground state will change from 3B1to 'A1 at 0 = 130"). (c) Every calculation predicts that the energy of the 3B1and 'B1 states is a very weak function of 0 for 8 > 120". Indeed, in our best calculation (VB 111) the energy separation 3Bl(1800) - 3B1(1380) is 0.19 eV, and 'B1( 1SO') - 'B1( 148 ") is only 0.08 e\?. (d) Every calculation (with the exception of VB I) predicts that the energy separation between the jB1 and ]Al states increases monotonically with increasing H C H angle. In VB 111 the difl'erence is 1.77 eV at 8 = 138". From the preceding observations we conclude that our calculations predict (i) the order of the states is 3B1 < 'A1 < 'B1 for values of the H C H angle between 90 and 180 ". (ii) Because of the very flat potential curves of the 3B1and 'B1 states, the predicted equilibrium angles of 138 and 148", respectively, could be changed significantly with minor changes in the atomic bases. The calculated equilibrium angle of 108" for the 'A1 state is more reliable. Thus our results are in over-all agreement with Herzberg's spectroscopic conclusion : j 3B1 (-180") < 'A1(103') < 'B1(1400). With regard to detailed differences it seems reasonable to conclude that the theoretical and experimental values are at about the same level of conclusiveness and thus these differences remain as present uncertainties. (iii) In view of (ii) we must give 1.77-2.10 eV as the range for the probable energy of the 3Bi-1A1separation. E. Heat of Atomization. The heats of atomization,
Cottrell'"" estimates x -5.2 eV while TrotmanDickensonlob gives x -4.8 eV. We therefore have the estimates AH(3B1) E -8.7 or -8.3 eV, and we take these as being indicative of the probable magnitude of the desired quantity. We present in Table XI the heats of atomization which we have calculated and for comparison the results of Foster and Boys.' The energy of the 3P state of carbon in our atomic basis is -37.6805 Hartree units.j We calculate the 'D energy to be -- 1.58 eV for the 3P-1D multiplet separation which is to be compared with the experimental value" of - 1.26 eV. TlLe data presented by Foster and Boys7 yields -2.01 eV for this splitting. O n the whole our best VB wave functions agree fairly well with the Foster and Boys results, and if we take -8.47 eV as a reasonable comproiimise between the two experimental estimates, the calculations yield respectively 75 and 7 6 x of the heat of reaction. C. Spectroscopy. (1) Experimental. HerzbergY has obtained the absorption spectra of CH2 in the vacuum ultraviolet, visible, and near-ultraviolet. He finds (a) a band at 8.76 eV, the structure of which is typical of a I: +-+ 2 transition (Herzberg assigns this as d "ir P - +- 3Zgand concludes that both states involved in the transition are linear or very nearly so); (b) a manylined spectrum between 1.305 and 2.25 eV (In this region bands are found at 1.513, 1.695, and 1.898 eV. These "red bands" are assigned rather definitely to the transition 'B1 'Al.); (c) a series of very weak bands, between 3.54 and 3.87 eV, which are favored by the conditions under which the red bands are produced (Herzberg tentatively assigns these to the transition 'A1 + 'Ai). 2. Theoretical Prediction of Triplet-Triplet Transitions. From Figure 3 we see that VB I predicts the lowest triplet to be "B1 with an equilibrium angle of 130" and that two states, "A? and :]A1, are accessible cia electric dipole transition from this 'B1state. The vertical excitation energies are
(7) J. M . Foster and S . F. Boys, Rer. Mod. P h j s., 26, 716 (1957). (8) G. Herzberg, Proc. Roy. SOC.(London), A262, 291 (1961); A295, 107 (1966).
Journal of the American Cliemicul Society
91:4
AH(8B1) = x(eV) - 3.47 eV
-
(9) G. Herzberg, "Spectra of Diatomic Molecules," 2nd ed, D. Van Nostrand Co., New York, N. Y., 1950. (10) (a) T. L. Cottrell, "The Strengths of Chemical Bonds," 2nd cd, Butterworth & Co., Ltd., London, 1958; (b) A. F. Trotman-Dickenson, Ann. Rept., 55, 36 (1958). (11) C. Moore, "Atomic
Energy Levels," Vol. 1, Nationnl Bureau of Standards Circular No. 467, U. S . Government Printing Office,
Washington, D. C.
Februury 12, 1969
815 Table X. Equilibrium Angles and Energies for Various Methylene Wave Functions VBa _____. Doubly occ Doubly occ Doubly occ 1s and 2s 1s only Is, scaled H's
-
7-
Om,,, deg Energye Om,,,, deg Energy Om,,, deg Energy
'Bi 1.41
3B1
135 -38 665 95 -38 157 130 -38 720
148
- 38.758 104 -38.790 132 -38.840
148 -38.833 108 - 38.864 138 -38.915
MOh
Foster and Boysd
Exptc
154 -38.822 111 -38 843 133 -38.893
140
;t
...
15
132 -38.808 90 -38 865 129 -38 904
102.4
180
The 'A1 state used is constructed from a variation calalculation over the two determinants defined by a C-H distance taken as 2.00 au. the configurations ( lal)2(2al)2(lb~)z(3a1)2 and (lal)z(2al)z(lb~)z~1bl)"; the 'B1 and states are constructed from the configuration ( I a1)2(2a,)2(lb1j2(3al)~(lb2j'formed by exciting an electron from the 3al MO to the lbz virtual orbital. The experimentai data are from G. Herzberg, ~ 1.95 au. C H distance for 'B1 and 3BIis 2.1 1 au; ref 8. This reference lists the experimental C H distances as 'A1 = 2.10, 'B1 = 1.98, 3 B = for 'A1 it is 2.21 au: J. M.Foster and S. F. Boys, Rev. Mod. Phys., 32, 305 (1960). Energy in Hartree units.
Table XI. Heats of Reactions from Various Methylene Wave Functions r-
Reaction A B
c
Doubly occ 1s and 2s
Doubly occ Is only
-1.06 -3.67 -1.17
-4.33 -4.57 -3.70
--
VB ___ Doubly occ Is, scaled H MO*2 -6.36 -6.58 -5.74
Foster and Boy9
-5.77 -6.01 -5.44
Expt
-6.42 -7.37 -5.83
-8.67, -8.27 ?
,
a The 'A, state constructed by a variational calculation over the two determinants defined by the configurations (~a~)2(2al)~(lbz)2(3al)z and (la1)2(2al)2(lbz)2(1bl)~ was used. The lB1 and 3B1states where constructed from the configuration resulting from a single excitaticn out of the 3al MO to the 1bl virtual orbital. b J. M. Foster and S. F. Boys, Reu. Mod. Phys., 32,305 (1960).
3A2+- 3B1 6.6 eV ( z polarization) "I
Table XII. Vertical Transition Energies for 'Ai
'BI
ecgul:*: deg
Calculation
AEa
VB-doubly occ 1s and 2s VB-doubly occ 1s only VB-doubly occ 1s and scaled H MOO MO* * Exps
3.41
95
1.44
104
1.52
108
0.98 1.11 1.51 1.70 1.90
107 111 102.4
+ 3B1 9.8 eV ( y polarization)
where the polarization of the transition is indicated relative t o the coordinate system defined in Figure 1 (we note that the experimental transition should be z polarized). In the linear configuration the three lowest states of 3A2symmetry correlate with the %rTTg,32u+, and 3Au states which in this calculation lie respectively 10.3, 12.8, and 13.4 eV above the 38,- state. The three lowest states of 3A1symmetry correlate with the 37r,, 3A6, and 32g+ states of linear methylene, which in this calcuiation lie respectively 6.85, 26.9, and 28.5 eV above the 32,- state. Since excitation of a carbon 2p electron to a 3s, 3p, or 3d orbital requires approximately 7.5, 8.5, and 9.6 eV,respectively, we see that the inclusion of 3s, 3p, and 3d orbitals in our atomic basis set is necessary for an adequate representation of the (predominately Rydberg) 3Anand 3A1 states. The probable Rydberg character of the excited triplets, rapid variation of the energy us. angle curves for the excited triplets, and the flatness of the 3B1 curve together result in a large uncertainty in the prediction of the triplet-triplet transition energy. 3. Theoretical Prediction of Singlet-Singlet Transitions. The longest wavelength singlet-singlet transition is in ail probability 'B1 + 'A1. Vertical transition energies as predicted by our calculations are presented in Table XII. Also presented is the equilibrium angle of the 'A1 state from which the transition originated. The adequacy with which our valence state represents the two singlet states involved in the transition is illustrated by the rather good agreement of our best calculation (VB111) with the experimental results. In view of the reasonable agreement with experiment, we calculated the oscillator strength for the VB I11 and MG* functions (MO* refers to the 'A1*state de-
-
a Energy in electron volts. Equilibrium angle of 'A1 state. The 'A1 stateis(la1)2(2a1)2(lb2)2(2a1)~; the 'B1 arises fromtheconfiguration (1a1)2(2a1)~(lb2)z(3al)1(lbi)1.The 'A1 state results from a variation calculation over the configurations (la1)2(2a1)2(1bl)z(3a1j and ( la1)2(2a1)2( 1by)2( 1b1)2.
fined by eq 4). We take as our definitionI2
where m and e are respectively the electronic mass and charge, Y is the frequency of the transition, and ji is the transition moment. All units are in the mks system. If we measure I. in units of eao and energy in Hartree units we have
f('B1
+
'Ai) =
'/3(AE)pV2
where AE = E('B1) - E(''A1)
and
(12) J. C. Slater, "Quantum Theory of Atomic Structure," Voi. 1, McGraw-Hill Book Co., Inc., New York, N. Y.,1960, p 156.
Harrison, Allen
1 Electronic Structure of Methylene
816
We present the oscillator strength, as a function of angle, in Table XIII. No experimental data are available for comparison. We note that even though AE for the MO* is smaller than for the VB, the larger transition moment as calculated with the MO* function causes the molecular orbital oscillator strength t o be slightly larger than that calculated with the VB function. Table XIII.
80 100 120 140
I.00C
Methylene Oscillator Strengths
0.4000 0.3277 0.2438 0.1321
0.4964 0.4067 0.2989 0.1620
0.0958 0.0655 0.0383 0.0169
0.0931 0.0619 0.0335 0.0112
0.0102 0.0047 0.0015 0.0002
0.0153 0.0068 0.0020 0.0002
Units of eao. Hartree units. The 'A1 state was constructed by a variational calculation over the two determinants defined by the configurations ( la1)2(2a,)*(1b2)2(3a1)2 and ( lal)2(2al)z(1b2)2(1b1)2; the lB1 state was constructed from the configuration resulting from a single excitation out of the 3al MO to the lbl virtual orbital. d VB functions with doubly occupied 1s and scaled hydrogen.
80
100
120
e-
140
160
I80
Figure 6. Dipole moment cs. angle for various states of CHZ. VB-doubly occupied 1s and scaled hydrogens.
IV. Properties A. Dipole Moment. We take as our definition (in atomic units, ea,)
Q
= x, y,
z
with ria being the cy coordinate of the ith electron and R,, being the a coordinate of the j t h nucleus. The coordinate system is as shown in Figure 1. By symmetry only pz is nonzero. We present in Table XIV the dipole moments of the lA1, IB1, and 3B1states as a function of the H C H angle for both the VB I11 and MO* functions. The graphical representation of these l a t a is shown in Figures 6 and 7. When one calculates the dipole moment curves for the 'B1 and 3B1 states using the MO* functions, one finds that the curves are identical. It is readily verified that when the lB1 and 3B1 states are defined by eq 2 and 3 they yield identical expectation values for any one-electron operator. However, because the 'B1 and 3B1states are predicted in the MO* representation t o have different equilibrium angles, they are predicted Table XIV.
80 100 120 140 160 180
Dipole Moments of MethyleneR
0.9951 0.9041 0.7866 0.5617 0 2289 0.0000
0.4671 0.3957 0.3502 0.2844 0.1651
o.oooo
0.9654 0.8922 0.7919 0.5829 0.2411
0.6004 0.5142 0.4464 0.3584 0.2114
0.4227 0.3502 0.3003 0.2420 0.1441
0.4671 0.3912 0.3368 0.2711 0.1609
0 . m o.oO0o 0 . m 0 . m
(atomic units, eao,hydrogen side positive). This state was constructed by a variational calculation over the two determinants and (la1)%defined by the configurations (la1)2(2al)2(lbz)z(3al)~ (2al)2(lb2)2(lbl)2. These states were constructed from the configurations formed by a single excitation out of the 3al M O to the VB wave functions with doubly occupied Is lbi virtual orbital. and scaled hydrogen. a pi
Journal of the American Chemical Society
9I:4
h
80
100
120
140
160
180
@Figure 7. Dipole moment os. angle for various states of CH2. M O calculations.
t o have different dipole moments, If we linearly interpolate the data presented in Table XIV to the predicted equilibrium angles of the various states, we obtain for the VB 111 calculation 0.852, 0.300, and 0.248 au for the dipole moments of the 1A1(108c), 'B1(148'), and 3B1 (138") states, respectively, and for the MO* calculation 0.840, 0.200, and 0.397 au for the 'A1( 111 "), 'B1(m"), and 3B1(133c), respectively. The VB and MO agree in their prediction for the IA1 state but give conflicting predictions for the order of the 'B1and 3B1states. Of course the predictions of the VB TI1 calculation are preferred. The fact that the dipole moment curve for the ly3R1 states as calculated in the MO* representation lies between the 'B1 and 3B1 curves as calculated with the VB I11 function lends itself to an interesting interpreta-
1 February 12, 1969
817
VB I11 wave function. The trends in these data may be understood by considering the change in the electron distribution which occurs when a 3al electron is promoted t o a lbl orbital. By considering the form of the ~av(B1)= [PZ('B~)f 3Pd3BdI/4 3al orbital as discussed in section II.D, we see that this promotion (al --t bl) should result in a decrease in the That this is indeed the situation, at least for small electron density along the CZaxis and thus a decrease in angles, is seen by the similarity of the last column in (xz), an increase in the electron density along the Table XIV t o the MO*, 1,3B2results. The deviation y axis and thus an increase in (yz): and a decrease at intermediate angles may be attributed to the dein the electron density along the z axis and thus a decrease in the quality of the one-electron orbitals as the crease in (zz). We see that these expectations are molecule approaches the linear configuration in which it quantitatively borne out by the data in Table XV. would be an open-shell situation. The apparent agreeWe also note the curious difference between the second ment at B = 160" might be attributed t o the fact that moments of the 'B1 and 3B1 states. The 'Bl seems t o both quantities must be equal t o zero at 180". be more extended than the 3B1in the y direction but In order t o understand why the dipole moment of the 'A1 state is (at all angles) larger than the ~ B or I 3 B ~ slightly less extended in the x and z directions. The diamagnetic susceptibility, x, is a sum of two dipoles, we will utilize the MO* representation. Accontributions, xd,a diamagnetic (negative) term arising cordingly we form the difference A p = pu,('A1*) in first-order perturbation theory, and x p , a paraniagPA 3 B ~ ) netic (positive) term arising in second-order perturbation theory.
tion. If we suppose that the MO* result is a statistical mixture of the singlet and triplet dipole curves, then it should be representable as the average
x Using eq 2 or 3 with 4 we obtain A p = cos 2X (3a11x/3al)
From the definition of X in eq 4 we expect that as B varies from 90 to 180", X varies from something greater than 90-135". Therefore the cos 2X is negative for values of 6' less than 180" and zero at 0 = 180". From the discussion in section II.D, we concluded that the average position of an electron in the 3al orbital was along the C2 axis and on the - x side. Therefore the integral (3a11x)3al) is negative and Ap 2 0 for 90" < 0 < 180". A more physical argument is t o note that the dipole moment is the difference between the sum of the x coordinates of the nuclei and the average position of an electron along the x axis. Because the average position of an electron in the 3al orbital is on the --x axis, removing one of these electrons to form a B1 configuration causes the average position t o increase, thereby resulting in a smaller dipole moment for a B1 state. B. Second Moments of Charge Distribution and the Larmor Term in the Diamagnetic Susceptibility. We present in Table XV the expectation values of the operator :r CY
=
' 2 8i=1
= x, y , and z
for the lAl, lB1, and 3B1 states as represented by the
Table XV.
Xd
+
x p
We will consider only the diamagnetic contribution to x. In general the diamagnetic term is a second-order tensor which in emu units is 2=1
If we measure the integral in atomic units a o zand use as our unit of susceptibility the erg/(Gz mole), we have 8
xaSd = (1.18845 X 10-6)($)criar,8 - 6,,q,2;$)
(5)
i=1
The Larmor term
xL is defined as XL =
' 1 3 trace xcra 8
XL =
-0.7923 X l e 6 (rC/ICr12]rC/) i=l
= -6.3384
X 1k6(rZ)
+
+
with (r2) = (x2) (y') (zz). If we consider the 'A1 state at the calculated equilibrium angle, we have (with carbon as origin) ( P ) = 3.1151 au and therefore xL = -19.750 X 10-6 erg/(Gz mole), while the 'B1 state yields ( r z ) = 3.0879 au and from this xL = -19.57 X erg/(Gz mole). Unfortunately an experimental value for the diamagnetic susceptibility of free methylene is not known. Because of its fundamental importance in the interpretation of susceptibility data of organic compounds there have been many attempts at abstracting a sus-
Second Moments of Charge Distribution5 VB-Doubly Occupied 1s with Scaled Hydrogen 3J31
0, deg 80 100 120 140 160
=
( X 2)
*
1.19187 1.06808 0.94976 0.85477 0.79572
0.77641 0.77606 0.77572 0.77573 0.77610
.
1Bl
>
(Y*)
(X')
(22,
1.17379 1.28348 1.38829 1.46854 1.51452
1.17205 1.05691 0.94472 0.85365 0.79660
(Yz,
(22,
0.78347 0.78377 0.78298 0.78108 0.77883
1.17373 1.28351 1.38546 1.46080 1.50101
(X2)
1.32786 1.20812 1.08706 0.96799 0.82672
(Y2, 0.60174 0.61391 0.62334 0.75583 0.74525
(22)
-
~~
~~
1.18538 1.29743 1.39815 1.46881 1.50619
8 a
Coordinate system as in Figure 1.
(x2)
=
(+1cxi21$)/8. a=1
Harrison, Allen J Electronic Structure of Methylene
818
ceptibility value characteristic of the CH, group. It is 2.2001 rather remarkable that almost all measurements indicate 2.000 Ia vduc for x between - 11 x and - 1 2 X loF6. Since this experimental x includes xp, we see xL < x. Our result for xL is consistent with this inequality, but since X P is probably of the order of 0.5 X our com1.600L puter xL is probably too large. If we neglect the para1.400magnetic contribution t o the diamagnetic suscep1 tibility, we may estimate the diamagnetic anisotropy of !200t the CH2 group from the elements of xesd. With the z- I O O O L coordinate system defined by Figure I this tensor is C diagonal. If we interpolate the data in Table XV = 8001 to the equilibrium angle of the 'A1 state we obtain, u $ 600.using 5, ,yIzd = -18.59 X lodfi, xu: = -23.74 X 0 l O P , and xzzd= -16.90 X lo+, whereas for the
